Fair representation in the intersection of two matroids

TL;DR

This paper explores fair representation in the intersection of two matroids, proposing a conjecture on the existence of a set meeting each partition element proportionally, and proves it for the case of two partitions.

Contribution

It introduces a conjecture on fair representation in the intersection of two matroids and proves it for bipartitions, advancing understanding in combinatorial optimization.

Findings

Conjecture on fair representation in intersections of two matroids.

Proof of the conjecture for bipartitions.

Extension of matroid intersection theory.

Abstract

For a simplicial complex ${\mathcal C}$ denote by $\beta({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in {\mathcal C}$ meeting each $A_i$ in at least $\frac{|A_i|}{\beta({\mathcal C})}$ elements. We conjecture that a slightly weaker result is true for the intersections of two matroids: if ${\mathcal D}={\mathcal P} \cap {\mathcal Q}$, where ${\mathcal P},{\mathcal Q}$ are matroids on the same ground set $V$ and $\beta({\mathcal P}), \beta({\mathcal P}) \le k$, then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in {\mathcal D}$ meeting each $A_i$ in at least $(\frac{1}{k}-\frac{1}{|V|})|A_i|-1$ elements. We prove this for a partition into two sets.